Solutions Of Nonlinear Hyperbolic Equations Research Articles

Multidimensional smoothness indicators for first-order Hamilton-Jacobi equations

The lack of smoothness is a common feature of weak solutions of nonlinear hyperbolic equations and is a crucial issue in their approximation. This has motivated several efforts to define appropriate indicators, based on the values of the approximate solutions, in order to detect the most troublesome regions of the domain. This information helps to adapt the approximation scheme in order to avoid spurious oscillations when using high-order schemes. In this paper we propose a genuinely multidimensional extension of the WENO procedure in order to overcome the limitations of indicators based on dimensional splitting. Our aim is to obtain new regularity indicators for problems in 2D and apply them to a class of “adaptive filtered” schemes for first order evolutive Hamilton-Jacobi equations. According to the usual procedure, filtered schemes are obtained by a simple coupling of a high-order scheme and a monotone scheme. The mixture is governed by a filter function F and by a switching parameter εn=εn(Δt,Δx)>0 which goes to 0 as (Δt,Δx) is going to 0. The adaptivity is related to the smoothness indicators and allows to tune automatically the switching parameter εjn in time and space. Several numerical tests on critical situations in 1D and 2D are presented and confirm the effectiveness of the proposed indicators and the efficiency of our scheme.

Approximate solution of nonlinear hyperbolic equations with homogeneous jump conditions

We present the error analysis of class of second order nonlinear hyperbolic interface problem where the spatial and time discretizations are based on finite element method and linearized backward difference scheme respectively.
 Both semi discrete and fully discrete schemes are analyzed with the assumption that the interface is arbitrary but smooth.
 Almost optimal convergence rate in \(H^1(\Omega)\)-norm is obtained.
 Examples are given to support the theoretical result.

Nonpoint Symmetry and Reduction of Nonlinear Evolution and Wave Type Equations

We study the symmetry reduction of nonlinear partial differential equations with two independent variables. We propose new ansätze reducing nonlinear evolution equations to system of ordinary differential equations. The ansätze are constructed by using operators of nonpoint classical and conditional symmetry. Then we find solution to nonlinear heat equation which cannot be obtained in the framework of the classical Lie approach. By using operators of Lie-Bäcklund symmetries we construct the solutions of nonlinear hyperbolic equations depending on arbitrary smooth function of one variable too.

Global and Blow-Up Solutions for Nonlinear Hyperbolic Equations with Initial-Boundary Conditions

We consider an initial-boundary value problem to a nonlinear string equations with linear damping term. It is proved that under suitable conditions the solution is global in time and the solution with a negative initial energy blows up in finite time.

Behavior of the solution of nonlinear hyperbolic equation in a domain with non-smooth boundary

In this paper the estimations of the solution and its derivative with respect to t of the initial boundary value problem for the second order hyperbolic type equation in a domain with non-smooth boundary are obtained. The question of smoothness of the generalized solution of the investigated problem is also studied.

Oscillations of nonlinear hyperbolic equations with functional arguments via Riccati method

This paper is devoted to the study of the oscillatory behavior of solutions of nonlinear hyperbolic equations with functional arguments by using integral averaging method and a generalized Riccati technique. First, we establish oscillation results for nonlinear hyperbolic equations by applying oscillation criteria for Riccati inequality. Secondly, we present interval oscillation results for nonlinear hyperbolic equations. These results improve and extend oscillation results of Cui and Xu [1].

Existence and stability of regularized shock solutions, with applications to rimming flows

This paper is concerned with regularization of shock solutions of nonlinear hyperbolic equations, i.e., introduction of a smoothing term with a coefficient ɛ, then taking the limit ɛ → 0. In addition to the classical use of regularization for eliminating physically meaningless solutions which always occur in non-regularized equations (e.g. waves of depression in gas dynamics), we show that it is also helpful for stability analysis. The general approach is illustrated by applying it to rimming flows, i.e., flows of a thin film of viscous liquid on the inside of a horizontal rotating cylinder, with or without surface tension (which plays the role of the regularizing effect). In the latter case, the spectrum of available linear eigenmodes appears to be continuous, but in the former, it is discrete and, most importantly, remains discrete in the limit of infinitesimally weak surface tension. The regularized (discrete) spectrum is fully determined by the point where the velocity of small perturbations vanishes, with the rest of the domain, including the shock region, being unimportant.

Strong convergence of approximate solutions for nonlinear hyperbolic equation without convexity

Schonbek [M.E. Schonbek, Convergence of solutions to nonlinear dispersive equations, Comm. Partial Differential Equations 7 (1982) 959–1000] obtained the strong convergence of uniform L loc p bounded approximate solutions to hyperbolic scalar equation under the assumption that the flux function is strictly convex. While in this paper, by constructing four families of Lax entropies, we succeed in dealing with the non-convexity with the aid of the well-known Bernstein–Weierstrass theorem, and obtaining the strong convergence of uniform L ∞ or L loc p bounded viscosity solutions for scalar conservation law without convexity.

On Doubly Periodic Solutions of Nonlinear Hyperbolic Equations of Higher Order

Abstract Unimprovable conditions of the existence and uniqueness of doubly periodic solutions are established for nonlinear hyperbolic equations of higher order with two independent variables.

Nonexistence of global solutions of nonlinear hyperbolic equation with material damping

Concerns with the nonexistence of global solutions to the initial boundary value problem for a nonlinear hyperbolic equation with material damping. Nonexitence theorems of global solutions to the above problem are proved by the energy method, Jensen inequality and the concavity method, respectively. As applications of our main results, three examples are given.

Numerical existence and uniqueness proof for solutions of nonlinear hyperbolic equations

We consider a numerical method to verify the existence and uniqueness of the solutions of nonlinear hyperbolic problems with guaranteed error bounds. Using a C 1 finite element solution and an inequality constituting a bound on the norm of the inverse operator of the linearized operator, we numerically construct a set of functions which satisfy the hypothesis of Banach's fixed point theorem for a continuous map on L p -space in a computer. We present detailed verification procedures and give some numerical examples.

On periodic in the plane solutions of nonlinear hyperbolic equations

On periodic in the plane solutions of nonlinear hyperbolic equations

Numerical verification of solutions for nonlinear hyperbolic equations

We consider a numerical method to verify the solutions for nonlinear hyperbolic problems with guaranteed error bounds in the one-space dimensional case. We present verification procedures and show some numerical examples.

Investigations of Solutions of Nonlinear Hyperbolic Equations with a Small Nonlinearity and Applications

The Cauchy problem for the nonlinear hyperbolic equation $L_\varepsilon u + \varepsilon ^p N_\varepsilon (t,x,D^\alpha u,| \alpha | \leq q) = f_\varepsilon (t,x)$ of order $m \geq 2$, $0 \leq q \leq m - 1$, is studied; $\varepsilon \in (0,1]$ is a small parameter. The problem possesses a solution on any given finite interval of time provided $\varepsilon $ is sufficiently small. Estimations for Sobolev and $L^\infty $ norms of solutions are derived. Applications to the stability problem for hyperbolic equations, and to justification of the nonlinear geometric optics method for spin glass models, are given.

On classical almost-periodic solutions of nonlinear hyperbolic equations

On classical almost-periodic solutions of nonlinear hyperbolic equations

A uniqueness result for measure-valued solutions of nonlinear hyperbolic equations

A uniqueness result for measure-valued solutions of nonlinear hyperbolic equations

Oscillations of the solutions of nonlinear hyperbolic equations of neutral type

In this paper nonlinear hyperbolic equations of neutral type of a given form are considered, with certain boundary conditions. Under certain constraints on the coefficients of the equation and the boundary conditions, sufficient conditions for oscillation of the solutions of the problems considered are obtained.

Local Error Estimates for Discontinuous Solutions of Nonlinear Hyperbolic Equations

Let $u(x,t)$ be the possibly discontinuous entropy solution of a nonlinear scalar conservation law with smooth initial data. Suppose $u_\varepsilon (x,t)$ is the solution of an approximate viscosity regularization, where $\varepsilon > 0$ is the small viscosity amplitude. It is shown that by post-processing the small viscosity approximation $u_\varepsilon $, pointwise values of u and its derivatives with an error as close to e as desired can be recovered. The analysis relies on the adjoint problem of the forward error equation, which in this case amounts to a backward linear transport equation with discontinuous coefficients. The novelty of our approach is to use a (generalized) E-condition of the forward problem in order to deduce a $W^{1,\infty } $-energy estimate for the discontinuous backward transport equation; this, in turn, leads to e-uniform estimate on moments of the error $u_\varepsilon - u$. The approach presented does not “follow the characteristics” and, therefore, applies mutatis mutandis to other approximate solutions such as E-difference schemes.

Reflection and interaction of progressing waves for semilinear wave equations

Recently, many works have been devoted to the discussion of the propagation and interaction of singularities for solutions of nonlinear hyperbolic equations. Among them the study of the interaction of nonlinear progressing waves is noticeable. First in [ 11, J. M. Bony discussed the behavior of’the interaction of two progressing waves. He showed that for semilinear hyperbolic equations of second order, when two progressing waves intersect transversally, no anomalous singularities are caused by the interaction, and for semilinear hyperbolic equations of higher order anomalous progressing waves appear issuing from the intersection. However, the interaction is much more complicated when three progressing waves intersect transversally with each other at one point. J. Rauch and M. Reed constructed an example showing that in the triple-intersection case, besides the linear-predicted singularities, a new progressing wave caused by interaction issuing from the intersection point propagates along the characteristic cone. Recently, R. Melrose and N. Ritter [3], J. M. Bony 141, and M. Beals [5] discussed the general triple-intersection of progressing waves for semilinear wave equations with two space variables by different methods. They pointed out that in the generalized case the singularities have the same structure as that in RauchhReed’s example. On the other hand, M. Beals and G. Metivier considered reflection of progressing waves for boundary value problems of semilinear hyperbolic equations. On the basis of these works, we study in this paper the interaction of two progressing waves with reflection on the boundary for boundary value problems of semilinear wave equations in two space variables. Our result shows that, in this case, beside the linear-predicted singularities there appears an anomalous reflected wave, which issues from the intersection

Decomposition solution of nonlinear hyperbolic equations

Some recent results in application of the decomposition method are discussed and a dissipative wave equation example is considered.